Real-rooted atlas

A137597 → reverse → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 - 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(-2 + 10\,x^{2} + 8\,x^{3} - 4\,x^{4} - n - 6\,n x - n x^{2} + 18\,n x^{3} + 12\,n x^{4} - 4\,n x^{5} - 2\,n^{2} x - 9\,n^{2} x^{2} - 8\,n^{2} x^{3} + 5\,n^{2} x^{4} + 6\,n^{2} x^{5} - n^{3} x^{2} - 4\,n^{3} x^{3} - 5\,n^{3} x^{4} - 2\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(2\,x^{2} + 4\,x^{3} - 2\,x^{5} + n x^{2} + 6\,n x^{3} + 8\,n x^{4} + n x^{5} - 2\,n x^{6} + n^{2} x^{3} + 4\,n^{2} x^{4} + 5\,n^{2} x^{5} + 2\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(3 - 5\,x^{2} + 6\,x^{3} + 2\,n + 8\,n x - n x^{2} - 10\,n x^{3} + 2\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-x^{2} + x^{3} + 2\,x^{4} - n x^{2} - 3\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21101526
31186660145
4128183396260868
51404051540235512185559
615478045601208014316613937930
717013651134045780928209006933120274565

All 15 rows

Technical details
Node id
n-6e2b6743fd2a183f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8f766a22b21a7253

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   4;
  2: 1,  10,   15;
  3: 1,  18,   66,    60;
  4: 1,  28,  183,   396,    260;
  5: 1,  40,  405,  1540,   2355,    1218;
  6: 1,  54,  780,  4560,  12080,   14316,    6139;
  7: 1,  70, 1365, 11340,  45780,   92820,   90069,   33120;
  8: 1,  88, 2226, 24920, 142205,  435540,  715505,  589448,  190323;
  9: 1, 108, 3438, 49896, 382515, 1646316, 4052503, 5602488, 4019580, 1159750;
%F Formula
The row polynomials satisfy (n+1+(2*n-2)*x)*P(n,x) = -(n+2+(2*n^2+6*n)*x+(n^3+9*n^2+n-10)*x^2+(4*n^3+8*n^2-18*n-8)*x^3+(5*n^3-5*n^2-12*n+4)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n+2)*x^2+(n^2+6*n+4)*x^3+(4*n^2+8*n)*x^4+(5*n^2+n-2)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (2*n+3+(2*n^2+8*n)*x+(6*n^2-n-5)*x^2+(4*n^2-10*n+6)*x^3)*P(n-1,x) - ((n+1)*x^2+(3*n-1)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 + (2*n^2 + 6*n)*x + (n^3 + 9*n^2 + n - 10)*x^2 + (4*n^3 + 8*n^2 - 18*n - 8)*x^3 + (5*n^3 - 5*n^2 - 12*n + 4)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n + 2)*x^2 + (n^2 + 6*n + 4)*x^3 + (4*n^2 + 8*n)*x^4 + (5*n^2 + n - 2)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (2*n + 3 + (2*n^2 + 8*n)*x + (6*n^2 - n - 5)*x^2 + (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] - ((n + 1)*x^2 + (3*n - 1)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n + 1 + (2*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 2 + (2*n^2 + 6*n)*x + (n^3 + 9*n^2 + n - 10)*x^2 + (4*n^3 + 8*n^2 - 18*n - 8)*x^3 + (5*n^3 - 5*n^2 - 12*n + 4)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n + 2)*x^2 + (n^2 + 6*n + 4)*x^3 + (4*n^2 + 8*n)*x^4 + (5*n^2 + n - 2)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (2*n + 3 + (2*n^2 + 8*n)*x + (6*n^2 - n - 5)*x^2 + (4*n^2 - 10*n + 6)*x^3)*v[i - 1] - ((n + 1)*x^2 + (3*n - 1)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n + 1 + (2*n - 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.